Unique determinations in inverse scattering problems with phaseless near-field measurements

Abstract

In this paper, we establish the unique determination results for several inverse acoustic scattering problems using the modulus of the near-field data. By utilizing the superpositions of point sources as the incident waves, we rigorously prove that the phaseless near-fields collected on an admissible surface can uniquely determine the location and shape of the obstacle as well as its boundary condition and the refractive index of a medium inclusion, respectively. We also establish the uniqueness in determining a locally rough surface from the phaseless near-field data due to superpositions of point sources. These are novel uniqueness results in inverse scattering with phaseless nearfield data.

Some recent developments in the unique determinations in phaseless inverse acoustic scattering theory

“…Toward establishing the uniqueness for exterior inverse scattering problems, superposing certain point sources has the capability of providing more information and overcoming the indispensability of incorporating a reference ball in the previous section. By introducing a general admissible surface/curve, along with the superposition of point sources, we proved that the bounded scatterer (impenetrable obstacle or medium inclusion) and the locally perturbed half-plane (locally rough surface) could be uniquely determined from the phaseless near-field data [52]. The uniqueness can be also established by the superposition of incident point sources and phaseless far-field data, see [44].…”

“…Theorem 3.2. [52] Assume that Γ and Σ are admissible surfaces with respect to Ω and G respectively, such that Ω ∩ G = ∅. For two scatterers D 1 and D 2 , suppose that respectively, then n 1 = n 2 .…”

“…We now consider the inverse scattering problem by the locally perturbed half-plane for incident point sources with limited-aperture phaseless near-field data. Analogous to Definition 3.1, the admissible curves need to be introduced [52]. (iii) Ω is non-resonant, i.e., k 2 is not a Dirichlet eigenvalue of −∆ in Ω; (iv) Λ ⊂ ∂Ω is a one-dimensional analytic manifold with non-vanishing measure.…”

“…Our studies on the uniqueness in phaseless inverse scattering problems are scattered in different papers [44,50,52,55] and thus they are loosely coupled in certain sense. Accordingly, this paper aims to present a unified and consistent framework for the problem formulations, technical treatment and the uniqueness results.…”

Generating geometric body shapes with electromagnetic source scattering techniques

“…We would like to mention in passing some relevant references [3,5,7,10,9,8,17,18,19] on the uniqueness and stability issues for inverse scattering problems that can provide the theoretical basis for the shape reconstruction in this subsection.…”